Real Numbers with Absolute Value form Normed Vector Space
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Theorem
Let $\R$ be the set of real numbers.
Let $\size {\, \cdot \,}$ be the absolute value.
Then $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space.
Proof
We have that:
By definition, $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed Spaces